Theorem cantnfres 9715

Description: The CNF function respects extensions of the domain to a larger ordinal. (Contributed by Mario Carneiro, 25-May-2015.)

Hypotheses

Ref	Expression
cantnfs.s	⊢ 𝑆 = dom (𝐴 CNF 𝐵)
cantnfs.a	⊢ (𝜑 → 𝐴 ∈ On)
cantnfs.b	⊢ (𝜑 → 𝐵 ∈ On)
cantnfrescl.d	⊢ (𝜑 → 𝐷 ∈ On)
cantnfrescl.b	⊢ (𝜑 → 𝐵 ⊆ 𝐷)
cantnfrescl.x	⊢ ((𝜑 ∧ 𝑛 ∈ (𝐷 ∖ 𝐵)) → 𝑋 = ∅)
cantnfrescl.a	⊢ (𝜑 → ∅ ∈ 𝐴)
cantnfrescl.t	⊢ 𝑇 = dom (𝐴 CNF 𝐷)
cantnfres.m	⊢ (𝜑 → (𝑛 ∈ 𝐵 ↦ 𝑋) ∈ 𝑆)

Assertion

Ref	Expression
cantnfres	⊢ (𝜑 → ((𝐴 CNF 𝐵)‘(𝑛 ∈ 𝐵 ↦ 𝑋)) = ((𝐴 CNF 𝐷)‘(𝑛 ∈ 𝐷 ↦ 𝑋)))

Distinct variable groups:   𝐵,𝑛   𝐷,𝑛   𝐴,𝑛   𝜑,𝑛

Allowed substitution hints:   𝑆(𝑛)   𝑇(𝑛)   𝑋(𝑛)

Proof of Theorem cantnfres

Dummy variables 𝑘 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cantnfrescl.d	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐷 ∈ On)
2		cantnfrescl.b	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ⊆ 𝐷)
3		cantnfrescl.x	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐷 ∖ 𝐵)) → 𝑋 = ∅)
4	1, 2, 3	extmptsuppeq 8243	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅) = ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))
5		oieq2 9544	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅) = ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅) → OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)) = OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅)))
6	4, 5	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)) = OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅)))
7	6	fveq1d 6918	. . . . . . . . . 10 ⊢ (𝜑 → (OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘) = (OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘))
8	7	3ad2ant1 1151	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)) ∧ 𝑧 ∈ On) → (OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘) = (OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘))
9	8	oveq2d 7467	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)) ∧ 𝑧 ∈ On) → (𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘)) = (𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘)))
10		suppssdm 8232	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅) ⊆ dom (𝑛 ∈ 𝐵 ↦ 𝑋)
11		eqid 2763	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ 𝐵 ↦ 𝑋) = (𝑛 ∈ 𝐵 ↦ 𝑋)
12	11	dmmptss 6261	. . . . . . . . . . . . . 14 ⊢ dom (𝑛 ∈ 𝐵 ↦ 𝑋) ⊆ 𝐵
13	12	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → dom (𝑛 ∈ 𝐵 ↦ 𝑋) ⊆ 𝐵)
14	10, 13	sstrid 3924	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅) ⊆ 𝐵)
15	14	3ad2ant1 1151	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)) ∧ 𝑧 ∈ On) → ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅) ⊆ 𝐵)
16		eqid 2763	. . . . . . . . . . . . . 14 ⊢ OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)) = OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))
17	16	oif 9561	. . . . . . . . . . . . 13 ⊢ OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)):dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))⟶((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)
18	17	ffvelcdmi 7115	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)) → (OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘) ∈ ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))
19	18	3ad2ant2 1152	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)) ∧ 𝑧 ∈ On) → (OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘) ∈ ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))
20	15, 19	sseldd 3914	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)) ∧ 𝑧 ∈ On) → (OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘) ∈ 𝐵)
21	20	fvresd 6936	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)) ∧ 𝑧 ∈ On) → (((𝑛 ∈ 𝐷 ↦ 𝑋) ↾ 𝐵)‘(OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘)) = ((𝑛 ∈ 𝐷 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘)))
22	2	3ad2ant1 1151	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)) ∧ 𝑧 ∈ On) → 𝐵 ⊆ 𝐷)
23	22	resmptd 6050	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)) ∧ 𝑧 ∈ On) → ((𝑛 ∈ 𝐷 ↦ 𝑋) ↾ 𝐵) = (𝑛 ∈ 𝐵 ↦ 𝑋))
24	23	fveq1d 6918	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)) ∧ 𝑧 ∈ On) → (((𝑛 ∈ 𝐷 ↦ 𝑋) ↾ 𝐵)‘(OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘)) = ((𝑛 ∈ 𝐵 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘)))
25	8	fveq2d 6920	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)) ∧ 𝑧 ∈ On) → ((𝑛 ∈ 𝐷 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘)) = ((𝑛 ∈ 𝐷 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘)))
26	21, 24, 25	3eqtr3d 2806	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)) ∧ 𝑧 ∈ On) → ((𝑛 ∈ 𝐵 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘)) = ((𝑛 ∈ 𝐷 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘)))
27	9, 26	oveq12d 7469	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)) ∧ 𝑧 ∈ On) → ((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐵 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘))) = ((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐷 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘))))
28	27	oveq1d 7466	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)) ∧ 𝑧 ∈ On) → (((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐵 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘))) +o 𝑧) = (((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐷 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘))) +o 𝑧))
29	28	mpoeq3dva 7528	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐵 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘))) +o 𝑧)) = (𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐷 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘))) +o 𝑧)))
30	6	dmeqd 5899	. . . . . 6 ⊢ (𝜑 → dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)) = dom OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅)))
31		eqid 2763	. . . . . 6 ⊢ On = On
32		mpoeq12 7524	. . . . . 6 ⊢ ((dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)) = dom OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅)) ∧ On = On) → (𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐷 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘))) +o 𝑧)) = (𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐷 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘))) +o 𝑧)))
33	30, 31, 32	sylancl 598	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐷 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘))) +o 𝑧)) = (𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐷 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘))) +o 𝑧)))
34	29, 33	eqtrd 2798	. . . 4 ⊢ (𝜑 → (𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐵 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘))) +o 𝑧)) = (𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐷 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘))) +o 𝑧)))
35		eqid 2763	. . . 4 ⊢ ∅ = ∅
36		seqomeq12 8502	. . . 4 ⊢ (((𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐵 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘))) +o 𝑧)) = (𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐷 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘))) +o 𝑧)) ∧ ∅ = ∅) → seqω((𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐵 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘))) +o 𝑧)), ∅) = seqω((𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐷 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘))) +o 𝑧)), ∅))
37	34, 35, 36	sylancl 598	. . 3 ⊢ (𝜑 → seqω((𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐵 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘))) +o 𝑧)), ∅) = seqω((𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐷 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘))) +o 𝑧)), ∅))
38	37, 30	fveq12d 6923	. 2 ⊢ (𝜑 → (seqω((𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐵 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘))) +o 𝑧)), ∅)‘dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))) = (seqω((𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐷 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘))) +o 𝑧)), ∅)‘dom OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))))
39		cantnfs.s	. . 3 ⊢ 𝑆 = dom (𝐴 CNF 𝐵)
40		cantnfs.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ On)
41		cantnfs.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ On)
42		cantnfres.m	. . 3 ⊢ (𝜑 → (𝑛 ∈ 𝐵 ↦ 𝑋) ∈ 𝑆)
43		eqid 2763	. . 3 ⊢ seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐵 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘))) +o 𝑧)), ∅) = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐵 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘))) +o 𝑧)), ∅)
44	39, 40, 41, 16, 42, 43	cantnfval2 9707	. 2 ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘(𝑛 ∈ 𝐵 ↦ 𝑋)) = (seqω((𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐵 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))‘𝑘))) +o 𝑧)), ∅)‘dom OrdIso( E , ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅))))
45		cantnfrescl.t	. . 3 ⊢ 𝑇 = dom (𝐴 CNF 𝐷)
46		eqid 2763	. . 3 ⊢ OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅)) = OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))
47		cantnfrescl.a	. . . . 5 ⊢ (𝜑 → ∅ ∈ 𝐴)
48	39, 40, 41, 1, 2, 3, 47, 45	cantnfrescl 9714	. . . 4 ⊢ (𝜑 → ((𝑛 ∈ 𝐵 ↦ 𝑋) ∈ 𝑆 ↔ (𝑛 ∈ 𝐷 ↦ 𝑋) ∈ 𝑇))
49	42, 48	mpbid 235	. . 3 ⊢ (𝜑 → (𝑛 ∈ 𝐷 ↦ 𝑋) ∈ 𝑇)
50		eqid 2763	. . 3 ⊢ seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐷 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘))) +o 𝑧)), ∅) = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐷 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘))) +o 𝑧)), ∅)
51	45, 40, 1, 46, 49, 50	cantnfval2 9707	. 2 ⊢ (𝜑 → ((𝐴 CNF 𝐷)‘(𝑛 ∈ 𝐷 ↦ 𝑋)) = (seqω((𝑘 ∈ dom OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴 ↑o (OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘)) ·o ((𝑛 ∈ 𝐷 ↦ 𝑋)‘(OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))‘𝑘))) +o 𝑧)), ∅)‘dom OrdIso( E , ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))))
52	38, 44, 51	3eqtr4d 2808	1 ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘(𝑛 ∈ 𝐵 ↦ 𝑋)) = ((𝐴 CNF 𝐷)‘(𝑛 ∈ 𝐷 ↦ 𝑋)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  Vcvv 3454   ⊆ wss 3900   ∖ cdif 4048  ∅c0 4281   ↦ cmpt 5192   E cep 5557  dom cdm 5658   ↾ cres 5660  Oncon0 6388  ‘cfv 6566  (class class class)co 7451   ∈ cmpo 7453   supp csupp 8214  seq_ωcseqom 8495   +_o coa 8511   ·_o comu 8512   ↑_o coe 8513  OrdIsocoi 9540   CNF ccnf 9699

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-isom 6575  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-1st 8035  df-2nd 8036  df-supp 8215  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-seqom 8496  df-1o 8514  df-oadd 8518  df-map 8887  df-en 9012  df-dom 9013  df-sdom 9014  df-fin 9015  df-fsupp 9392  df-oi 9541  df-cnf 9700

This theorem is used by:  cantnf2  44588